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Kannan fixed point theorem on generalized metric spaces. (English) Zbl 1161.54022
Let $X$ be a nonempty set and $d$ be a nonnegative symmetric function defined on $X^2$ such that $d(x,y)=0$ iff $x=y$ and $d(x,y)\le d(x,w)+d(w,z)+ d(z,y)$ for all $w,z$ in $X^{x,y}$(rectangular inequality). Then $(X,d)$ is called a generalized metric space and it is complete if every Cauchy sequence converges (the definition of Cauchy sequence is analogous to the usual case of metric spaces). The authors prove that if $T$ is a Kannan selfmap of $X$ (i.e., $T$ satisfies the inequality $d(Tx,Ty)\le h[d(x,Tx)+d(y,Ty)]$ for all $x,y$ in $X$ and for some $h$ with $0\le h \le 1)$, then there is a unique fixed point of $T$ in $X$. An analogous result concerning contraction maps on the above spaces was established by {\it A. Branciari} [Publ. Math. Debrecen 57, No. 1--2, 31--37 (2000; Zbl 0963.54031)].

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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